The fluid outside the turbulent region is either in irrotational motion (as in the case of a wake or a boundary layer), or nearly static (as in the case of a jet). Observations show that the instantaneous interface between the turbulent and nonturbulent fluid is very sharp.
The thickness of the interface must equal the size of the smallest scales in the flow, namely the Kolmogorov microscale.
Figure 33.1 Three types of free turbulent flows; (a) wake (b) jet and (c) shear layer [after P.K. Kundu and I.M. Cohen, Fluid Mechanics, Academic Press, 2002]
Derivation of Governing Equations for Turbulent Flow
We can write
From Eqs (33.3a) and (33.2), we obtain
Imagine a two-dimensional flow in which the turbulent components are independent of the z -direction. Eventually, Eq.(33.3b) tends to
On the basis of condition (33.4), it is postulated that if at an instant there is an increase in u' in the x -direction, it will be followed by an increase in v' in the negative y -direction. In other words, is non-zero and negative. (see Figure 33.2)
Invoking the concepts of eqn. (32.8) into the equations of motion eqn (33.1 a, b, c), we obtain expressions in terms of mean and fluctuating components. Now, forming time averages and considering the rules of averaging we discern the following. The terms which are linear, such as and vanish when they are averaged [from (32.6)]. The same is true for the mixed terms like ,u' or , v but the quadratic terms in the fluctuating components remain in the equations. After averaging, they form
Contd. from previous slide
If we perform the aforesaid exercise on the x-momentum equation, we obtain
using rules of time averages,
Introducing simplifications arising out of continuity Eq. (33.3a), we shall obtain.
In x direction,
In y direction,
In z direction,
Comments on the governing equation :
The left hand side of Eqs (33.5a)-(33.5c) are essentially similar to the steady-state Navier-Stokes equations if the velocity components u,v and w are replaced by,
The same argument holds good for the first two terms on the right hand side of Eqs (33.5a)-(33.5c).
However, the equations contain some additional terms which depend on turbulent fluctuations of the stream. These additional terms can be interpreted as components of a stress tensor.
Now, the resultant surface force per unit area due to these terms may be considered as
In x direction,
It can be said that the mean velocity components of turbulent flow satisfy the same Navier-Stokes equations of laminar flow. However, for the turbulent flow, the laminar stresses must be increased by additional stresses which are given by the stress tensor (33.7). These additional stresses are known as apparent stresses of turbulent flow or Reynolds stresses . Since turbulence is considered as eddying motion and the aforesaid additional stresses are added to the viscous stresses due to mean motion in order to explain the complete stress field, it is often said that the apparent stresses are caused by eddy viscosity . The total stresses are now
and so on. The apparent stresses are much larger than the viscous components, and the viscous stresses can even be dropped in many actual calculations .
Turbulent Boundary Layer Equations
The turbulent boundary layer equation together with the equation of continuity becomes
which can be integrated as , =constant
However, the wall shear stress in the vicinity ofthe laminar sublayer is estimated as
where Us is the fluid velocity at the edge of the sublayer. The flow in the sublayer is specified by a velocity scale (characteristic of this region).
as our velocity scale. Once ur is specified, the structure of the sub layer is specified. It has been confirmed experimentally that the turbulent intensity distributions are scaled with ur . For example, maximum value of the is always about . The relationship between ur and the Us can be determined from Eqs (33.11a) and (33.11b) as
Let us assume Now we can write
where is a proportionality constant (33.12a)
Hence, a non-dimensional coordinate may be defined as, which will help us estimating different zones in a turbulent flow. The thickness of laminar sublayer or viscous sublayer is considered to be n ≈ 5.
Turbulent effect starts in the zone of n > 5 and in a zone of 5< n < 70, laminar and turbulent motions coexist. This domain is termed as buffer zone. Turbulent effects far outweight the laminar effect in the zone beyond n = 70 and this regime is termed as turbulent core .
where the channel is assumed to have a height 2h and is the distance measured from the centreline of the channel (= h - y). Figure 33.1 explains such variation of turbulent stress.
Shear Stress Models
In analogy with the coefficient of viscosity for laminar flow, J. Boussinesq introduced a mixing coefficient μr for the Reynolds stress term, defined as
Using μr the shearing stresses can be written as
such that the equation
may be written as
The term vt is known as eddy viscosity and the model is known as eddy viscosity model .
Unfortunately the value of vt is not known. The term vt is a property of the fluid whereas νt is attributed to random fluctuations and is not a property of the fluid. However, it is necessary to find out empirical relations between vt, and the mean velocity. The following section discusses relation between the aforesaid apparent or eddy viscosity and the mean velocity components
Prandtl's Mixing Length Hypothesis
Fig. 33.4 One-dimensional parallel flow and Prandtl's mixing length hypothesis
The above expression is obtained by expanding the function in a Taylor series and neglecting all higher order terms and higher order derivatives. l is a small length scale known as Prandtl's mixing length . Prandtl proposed that the transverse displacement of any fluid particle is, on an average, 'l' .
Consider another lump of fluid with a negative value of v'. This is arriving at y1 from ( y1 + 1). If this lump retains its original momentum, its mean velocity at the current lamina y1 will be somewhat more than the original mean velocity of y1. This difference is given by
along with the condition that the moment at which u' is positive, v' is more likely to be negative and conversely when u' is negative. Possibly, we can write at this stage
where C1 and C2 are different proportionality constants. However, the constant C2 can now be included in still unknown mixing length and Eg. (33.18) may be rewritten as
(33.20a)where the apparent viscosity may be expressed as
(33.20b)and the apparent kinematic viscosity is given by
where Y is the distance from the wall and X is known as von Karman constant (≈ 0.4 ).